I have the following puzzle and I want to check if the answers that I gave are correct. You can check also my reasoning behind the answers. The puzzle is:
In a perfect world two people are either friends or not friends. In this perfect world the population is exactly 6 people.
In this perfect world a group of $n$ people is called completely friends when each person in the group is friends with every other person in the group.
In this perfect world a group of $n$ people is called completely not friends when each person in the group is not friends with every other person in the group.
Which of the below statements are always true?
We can always find a group of 3 which is either completely friends or completely not friends.
If we have a group of 4 which are completely friends then the number of friendships is higher than the number of not friendships. (A friendship is when two people are friends. A not friendship is when two people are not friends.)
If we have a group of 3 which are completely not friends, there is no way for each person to be friend with exactly two others.
If we know that each two people have at least one common friend, then this means that we have a group of 4 which are completely friends.
There is no way for each two people to have exactly one common friend (from the other four).
There is no way for each two people to have exactly one common not friend (from the other four).